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Theorem f1omo 48691
Description: There is at most one element in the function value of a constant function whose output is 1o. (An artifact of our function value definition.) Proof could be significantly shortened by fvconstdomi 48690 assuming ax-un 7754 (see f1omoALT 48692). (Contributed by Zhi Wang, 19-Sep-2024.)
Hypothesis
Ref Expression
f1omo.1 (𝜑𝐹 = (𝐴 × {1o}))
Assertion
Ref Expression
f1omo (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹𝑋))
Distinct variable groups:   𝑦,𝐴   𝑦,𝐹   𝑦,𝑋   𝜑,𝑦

Proof of Theorem f1omo
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 1oex 8515 . . . 4 1o ∈ V
2 eqid 2735 . . . 4 ((𝐴 × {1o})‘𝑋) = ((𝐴 × {1o})‘𝑋)
31, 2fvconst0ci 48689 . . 3 (((𝐴 × {1o})‘𝑋) = ∅ ∨ ((𝐴 × {1o})‘𝑋) = 1o)
4 mo0 48662 . . . 4 (((𝐴 × {1o})‘𝑋) = ∅ → ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋))
5 el1o 8532 . . . . . . . 8 (𝑦 ∈ 1o𝑦 = ∅)
6 el1o 8532 . . . . . . . 8 (𝑥 ∈ 1o𝑥 = ∅)
7 eqtr3 2761 . . . . . . . 8 ((𝑦 = ∅ ∧ 𝑥 = ∅) → 𝑦 = 𝑥)
85, 6, 7syl2anb 598 . . . . . . 7 ((𝑦 ∈ 1o𝑥 ∈ 1o) → 𝑦 = 𝑥)
98gen2 1793 . . . . . 6 𝑦𝑥((𝑦 ∈ 1o𝑥 ∈ 1o) → 𝑦 = 𝑥)
10 eleq1w 2822 . . . . . . 7 (𝑦 = 𝑥 → (𝑦 ∈ 1o𝑥 ∈ 1o))
1110mo4 2564 . . . . . 6 (∃*𝑦 𝑦 ∈ 1o ↔ ∀𝑦𝑥((𝑦 ∈ 1o𝑥 ∈ 1o) → 𝑦 = 𝑥))
129, 11mpbir 231 . . . . 5 ∃*𝑦 𝑦 ∈ 1o
13 eleq2w2 2731 . . . . . 6 (((𝐴 × {1o})‘𝑋) = 1o → (𝑦 ∈ ((𝐴 × {1o})‘𝑋) ↔ 𝑦 ∈ 1o))
1413mobidv 2547 . . . . 5 (((𝐴 × {1o})‘𝑋) = 1o → (∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋) ↔ ∃*𝑦 𝑦 ∈ 1o))
1512, 14mpbiri 258 . . . 4 (((𝐴 × {1o})‘𝑋) = 1o → ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋))
164, 15jaoi 857 . . 3 ((((𝐴 × {1o})‘𝑋) = ∅ ∨ ((𝐴 × {1o})‘𝑋) = 1o) → ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋))
173, 16ax-mp 5 . 2 ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋)
18 f1omo.1 . . . . 5 (𝜑𝐹 = (𝐴 × {1o}))
1918fveq1d 6909 . . . 4 (𝜑 → (𝐹𝑋) = ((𝐴 × {1o})‘𝑋))
2019eleq2d 2825 . . 3 (𝜑 → (𝑦 ∈ (𝐹𝑋) ↔ 𝑦 ∈ ((𝐴 × {1o})‘𝑋)))
2120mobidv 2547 . 2 (𝜑 → (∃*𝑦 𝑦 ∈ (𝐹𝑋) ↔ ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋)))
2217, 21mpbiri 258 1 (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 847  wal 1535   = wceq 1537  wcel 2106  ∃*wmo 2536  c0 4339  {csn 4631   × cxp 5687  cfv 6563  1oc1o 8498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-1o 8505
This theorem is referenced by:  indthinc  48853  prsthinc  48855
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